To find the quotient of the complex numbers [tex]\( (3 + 3i) \div (5 + 4i) \)[/tex], we follow these steps:
1. Define the complex numbers:
Let [tex]\( z_1 = 3 + 3i \)[/tex] and [tex]\( z_2 = 5 + 4i \)[/tex].
2. Express the quotient in terms of multiplication:
When dividing complex numbers, we multiply by the conjugate of the denominator. The conjugate of [tex]\( z_2 = 5 + 4i \)[/tex] is [tex]\( \overline{z_2} = 5 - 4i \)[/tex]. Thus,
[tex]\[
\frac{z_1}{z_2} = \frac{3 + 3i}{5 + 4i} \times \frac{5 - 4i}{5 - 4i}
\][/tex]
3. Perform the multiplication:
Calculate the numerator:
[tex]\[
(3 + 3i)(5 - 4i) = 15 - 12i + 15i - 12i^2 = 15 + 3i + 12 = 27 + 3i
\][/tex]
since [tex]\(i^2 = -1\)[/tex].
Calculate the denominator:
[tex]\[
(5 + 4i)(5 - 4i) = 25 - 20i + 20i - 16i^2 = 25 + 16 = 41
\][/tex]
since [tex]\(i^2 = -1\)[/tex].
4. Combine the results:
Now we have:
[tex]\[
\frac{27 + 3i}{41} = \frac{27}{41} + \frac{3}{41}i
\][/tex]
Thus, the quotient of the complex numbers [tex]\( (3+3i) \div (5+4i) \)[/tex] is:
[tex]\[
\boxed{\frac{27}{41} + \frac{3}{41} i}
\][/tex]
The correct answer is [tex]\( \text{B. } \frac{27}{41} + \frac{3}{41}i \)[/tex].
